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Doubly Linked Block Association Schemes K.K International Journal of Innovative Research in Engineering & Management (IJIREM) ISSN: 2350-0557, Volume-2, Issue-6, November-2015 Doubly Linked Block Association Schemes K.K. Singh Meitei Department of Statistics, Manipur University, Imphal, India Email: [email protected] treatments occurs λ* times, guarantees that any two blocks of the ABSTRTACT * The inception of the Singly Linked Block (SLB) association dual have λ treatments in common. The dual of a BIBD are not scheme was known to due to Shrikhande [1], defining the necessarily always partially balanced. From the literature of Shrikhande [6] it is learnt that the dual of a BIBD (v*, b*, r*, k*, association relationship of two blocks based on their intersection * * * * * number of treatment which must be either 0 or 1. In the present λ ) will be partially balanced if λ =1 or r =k, k =k-2, λ =2 work, starting from a given block design in which any two where k is the block-size of the dual design. blocks intersect at either 1 or 2 treatment(s), a new 2-class * association scheme to be named “Doubly Linked Block For the case λ =1, Shrikhande [6] and Roy [7] independently Association Scheme” is introduced and also some combinatorial showed that such design is two associate classes with the first approaches to its existence are laid down. type parameters and the second type parameters, Keywords * * * * * * * * * * * v= r (r k - r +1)/k , b=r k - r +1=v , r=k , k=r , λ1=1, λ2=0; Singly Linked Block (SLB) association scheme, λ*-linked n =k*(r*-1), n = (r*-k*)(k*-1)(r*-1)/k*, =(r*-2)+(k*-1)2, = blocks, affine α-resolvable BIB design. 1 2 . 1. INTRODUCTION Partially Balanced Incomplete Block (PBIB) designs were first This dual design under the case λ*=1 , is said to belong to the introduced by Bose and Nair [2], in order to exhaust the strictly singly linked blocks (SLB). The association scheme on which the restricted condition that the concurrences of every two design bases, is given by the designs themselves, since any two treatment should be same, as an extension of Balanced treatments are first associate when and only when they occur in Incomplete Block (BIB) design. Nair and Rao [3] generalized the same block. A certain overlap occurs between this type and the definition of PBIB designs, so as to include, as special the triangular type if k*=2, λ*=1. The design is the BIB design of cases, the cubic and other higher dimensional lattices. For Fisher and Yates [8] with parameters v*=n, b*=n(n-1)/2, r*=n-1, precise and comprehensive analysis of the Partially Balanced k*=2, λ*=1. The dual of this BIB design is the SLB type design Incomplete Block (PBIB) designs, later on it was realized that with parameters v=n(n-1)/2, b=n, r=2, k=n-1, λ1=1,λ2=0, n1=2n- those designs should base on a abstract relationship 4, =n-1, =4. configuration, so called Association Scheme. Further, introducing the concept of association schemes and basing the For the case λ*=2 , the parameters of the dual i.e. PBIBD are definition PBIB designs on these association schemes, the v=k*(k*-1)/2, b=(k*-1)(k*-2)/2, r=(k*-2), k=k*, λ =1,λ =2, contributions to the existence of plan of Group Divisible, 1 2 Triangular, Latin Square, Cyclic and Singly Linked Block n1=2k*-4, =k*-2, =4. This design comes under the (SLB) association schemes and that of plan of designs based on triangular design. We may call it Triangular doubly linked blocks them with the specific range of parameters r≤10 and k≤10 due as any two blocks interact at two common points, as shown in to Bose and Shimamoto [4] are available. design, reference number T44, page 243, Clathworthy [9]. Among all the association schemes, it is hereby mentioned a Let D* be a BIB design (v*, b*, r*, k*, λ*=1). The dual of the few of them viz. Group Divisible, Triangular, Latin Square, given design is a Singly Linked Block design D, say with Rectangular, Cyclic, Singly Linked Block (SLB) association parameters v=b*, b=v*, r=k*, k=r*. Two of any new treatments schemes and others. SLB association scheme is defined on the are first associates if the corresponding blocks in D* have a existence of a BIB design with the pairwise balancing common treatment and second associates if the corresponding parameter λ=1 . The existence of a SLB scheme is guarantee blocks of D have no common treatment. Shrikhande [1] has only when a BIB design with λ=1 exists. The inception of the shown that this relation satisfies the conditions of the association SLB association was known to due to Shrikhande [1], defining scheme for claiming it to be an association scheme. In general it the association relationship of two blocks based on their is known as Singly Linked Block association scheme. interaction number of treatment, which is either 0 or 1. Youden [5] proposed that the dual of a BIB design having the Given v treatments, viz., 1, 2, … , v, a relation satisfying the parameters v*, b*, r*, k*, λ* are so called “λ*-linked blocks” on the following conditions is said to be an association scheme with m reason that every pair of blocks of the dual design has λ* classes: * treatments in common. When λ =1(2), the dual design is called (a) Any two treatments are either 1st, 2nd, … , or m-th associates, Singly(Doubly) Linked Blocks. For “Singly(Doubly) Linked the relation of association being symmetrical, i.e., if the Blocks”, any two blocks of the given design have one(two) treatment α is an i-th associate of the treatment β, then β is also treatment(s) in common. It is known to all that the dual of any an i-th associate of the treatment α. balanced incomplete block design in which any pair of 101 Doubly Linked Block Association Schemes (b) Each treatment has ni i-th associates, the number ni being completed from the assumption (ii) that any two blocks having independent of α. one(two) treatment(s) in common, have exactly t(s) blocks each of which intersects with each of the two blocks exactly one I If any two treatments are i-th associates, then the number of treatment in common. treatments which are the j-th associates of α and k-th associates of β is and is independent of the pair of i-th associates α and For an illustration of the 2.1 Proposition, let us take the Serial No. 4.4.1 of the BIB design, Dey [11]. β. The numbers v, n , (i, j, k=1, 2, …, m), are the parameters i 2.1 Example: The BIB design has the parameters v=2t, of the association scheme. They hold good the following b=4t-2, r=2t-1, k=t, λ=1.Taking t=3, the parameters of the BIB design become v*=6, b*=10, r*=5, k*=3, λ*=1. The following is a relations, Bose and Nair [2]: =v-1, =nj-1 plan of the BIB design with the parameters v*=6, b*=10, r*=5, k*=3, λ*=2, where 1, 2, … , 6 represent the treatments and if i=j; = nj otherwise, and ni = nj = nk . parentheses indicate the blocks. B1=(1,2,3), B2=(1,2,4), B3=(1,3,5), B4=(1,4,6), B5=(1,5,6), B6=(2,3,6), B7=(2,4,5), B8=(2,5,6), B9=(3,4,6), B10=(3,4,5). Then the parameters of the 2. DOUBLY LINKED BLOCK DLB association scheme are v=10, n1=6, n2=3, =3, =0. ASSOCIATION SCHEME In the following it is being introduced the Doubly Linked Block 2.1 Theorem association scheme, starting from a BIBD with pairwise If there exists an affine α(>2) – resolvable BIB design with balancing parameter value 2. Consider a BIB design D with parameters v*, b*, r*, k*, λ*=2 such that parameters v, b=βt, r=αt, k, λ=2, q 1 =1, q 2 =2, then there exists a (i) any two blocks intersect either one or two treatment(s) Doubly Linked Block (DLB) association scheme with parameters in common, 2 0 n 1 =β-1, n 2 =β(t-1), P 1 = and (ii) any block has ni blocks which intersect i (=1,2) treatment(s) in common and no other blocks, 0 (t 1) (iii) any two blocks having one(two) treatment(s) in P = 0 1 . common, have exactly t(s) blocks each of which 2 1 t 2 intersects with each of the two blocks exactly one treatment in common. Proof: Consider an affine α-resolvable BIB design, D, with Define two block numbers of D to be first associates if they parameters v, b=βt, r=αt, k, λ=2, q 1 =1, q 2 =2, (b>v). Then the b have exactly one treatment in common and second associates if blocks can be divided into t sets, each of β blocks such that each they have exactly two treatments in common. Then this treatment occurs α times in each set of blocks. Further any two association rule is an association scheme with two classes, which blocks belonging to the same set contain q (i.e. one) treatment may be called Doubly Linked Block (DLB) association scheme. 1 The existence of the association scheme can be insured only in common and any two blocks belonging to the different sets when the BIBD has the concurrence parameter value 2 and its contain q (i.e.
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